Mathematization as a Component of the Concept of Ratio-as-Measure: A Study of Prospective Elementary Teachers.

Simon, Martin A.; Blume, Glendon W.

Research has demonstrated that people have difficulties recognizing ratio relationships involving multiplication. This paper is divided into four sections. The first section describes a study that was part of the three-year study of the mathematical and pedagogical development of prospective elementary teachers. The classroom teaching experiment involving 26 prospective elementary teachers employed constructivist teaching methodology to teach the concept of ratio-as-measure. Instruction was divided into two units. The first involved the development of understanding of the area of a rectangle as a multiplicative relationship between the sides. The second was a unit on the mathematical modeling of the concept of slope in order to learn to model situations with ratio. Researchers' analyses of videotapes of the classroom session led to the development of models of students' understanding. The second section presents and discusses three unanticipated difficulties encountered by the students in the exploration and discussion of slope: the notion of negative slopes, an inability to understand the relationship between the mathematical model and the situation that it models, and the inability to explore the validity of two methods presented to represent slope. The third section discusses the relationship between the students ability to understand the concept of ratio-as-measure and the larger concept of students' ability to understand mathematization, that is the process of mathematical modeling. In conclusion, mathematization of real world situations is a building block of the concept of ratio-as-measure. A suggestion is made for the development of mathematization throughout the mathematics curriculum. (18 references) (MDH)